Finite Group Databasebeta

← #27279325

Subgroups — #27279325

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285270C211yesmaximal, characteristic
2#20283459C211yesmaximal, characteristic
3#20275658C211yesmaximal, characteristic, complemented
4#20273119C211yesmaximal, characteristic, complemented
5#20272556C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#1502C211yesmaximal, characteristic
8C2×#13133224—21no—
9#16811180C2211yescharacteristic
10#16811019C411yescharacteristic, complemented
11#16810995C411yescharacteristic, complemented
12#16810975C411yescharacteristic, complemented
13#16810969C411yescharacteristic, complemented
14#16810850—22no—
15C4×#11719923—22no—
16#16810701—21no—
17#16810695—22no—
18D4×#88322—21no—
19C2×C4⋊#88322C2211yescharacteristic
20#16810590—21no—
21C4×#11719917C2211yescharacteristic
22C2×C4×#88322C2211yescharacteristic
23#16810383—21no—
24#16810383—21no—
25#16810355—21no—
26#16810355—21no—
27C53×#532C2211yescharacteristic
28C53×#431C2211yescharacteristic
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#327—22no—
31C2×#11719923—22no—
32#13133224—42no—
33#13133224—42no—
34C2×#11719917—22no—
35C2×#11719917D411yescharacteristic
36C2×#11719917D411yescharacteristic
37C22×#88322—22no—
38C22×#88322—21no—
39C22×#88322—21no—
40#13133217C2×C411yescharacteristic
41#13133217C2×C411yescharacteristic
42#13133209—21no—
43#13133196—21no—
44#13133196—21no—
45C4×Dic106—21no—
46C4×Dic106—21no—
47#13133176—21no—
48#13133148—22no—
49C4⋊#88322—21no—
50C4⋊#88322D411yescharacteristic
51C4⋊#88322D411yescharacteristic
52#13133140—22no—
53#13133138—22no—
54C22⋊#88322—22no—
55C22⋊#88322—22no—
56#13133128—21no—
57C4×#88322—21no—
58C4×#88322—21no—
59C4×#88322C2×C411yescharacteristic
60C4×#88322C2×C411yescharacteristic
61C2×#41×C53—21no—
62C4⋊Q8×C53C2×C411yescharacteristic
63#127×C53C2×C411yescharacteristic
64#124×C53—21no—
65#121×C53—21no—
66#121×C53D412yes—
67C4×D4×C53—21no—
68C4×D4×C53D412yes—
69C2×C4⋊C4×C53—21no—
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74C4×C8×C53—21no—
75#95×C53—21no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×D4×C53—22no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53C22⋊C412yes—
85C22×C4×C53—22no—
86C22×C4×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C2×Dic106—21no—
91C2×Dic106—21no—
92C2×C8×C53—41no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—41no—
96C4⋊C4×C53C22⋊C412yes—
97C22⋊C4×C53—22no—
98C22⋊C4×C53—22no—
99C42×C53—41no—
100C42×C53—41no—
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C22⋊Dic53—41no—
106#11719923—44no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—21no—
111#11719919—41no—
112#11719919—41no—
113C4×Dic53—41no—
114#11719917—21no—
115#11719917—21no—
116#11719917—21no—
117#11719917—44no—
118#11719917SD1612yescomplemented
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—42no—
126C2×#88322—44no—
127C2×#88322—41no—
128C2×#88322—41no—
129C23×C53—22no—
130C23×C53#12111yescharacteristic
131Q8×C53—41no—
132Q8×C53—41no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—41no—
137D4×C53—41no—
138D4×C53—41no—
139D4×C53—41no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—81no—
148C2×C4×C53—81no—
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53#12011yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12311yescharacteristic
155C2×C4×C53C2×SD1611yescharacteristic
156C2×C4×C53#13511yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C2×Dic53—41no—
160C2×Dic53—41no—
161Dic106—42no—
162Dic106—41no—
163Dic106—41no—
164C8×C53—81no—
165#88322—22no—
166#88322—22no—
167#88322—41no—
168#88322—41no—
169#88322—41no—
170#88322—41no—
171#9055—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53#38611yescharacteristic
178C22×C53#46111yescharacteristic
179C22×C53#46311yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53—82no—
185C4×C53C4×SD1611yescharacteristic
186C4×C53#44011yescharacteristic
187C4×C53#44311yescharacteristic
188C4×C53#44411yescharacteristic
189C4×C53#45011yescharacteristic
190C4×C53#45211yescharacteristic
191Dic53—81no—
192Dic53—81no—
193#3167—531no—
194#2935—531no—
195#1986—531no—
196#1502D5311yescharacteristic
197#1481—531no—
198#1279—531no—
199#1278—531no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—82no—
203C2×C53#187611yescharacteristic
204C2×C53#187711yescharacteristic
205C2×C53#188411yescharacteristic
206C2×#131—1061no—
207#533—531no—
208#532C2×D5311yescharacteristic
209#498—531no—
210#492—531no—
211#487—531no—
212#486—531no—
213#465—1062no—
214C4×SD16—1062no—
215#436—1061no—
216#435—1062no—
217C8×D4—1061no—
218#431C2×D5311yescharacteristic
219C2×C4⋊C8—531no—
220#408—1061no—
221C4×M16—531no—
222C2×C4×C8—531no—
223C4×C4⋊C4C2×D5311yescharacteristic
224#339—1061no—
225#339—1061no—
226#327—22no—
227Q8⋊C8—1061no—
228Q8⋊C8—1061no—
229C53#905511yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×SD16—1062no—
232#131—2122no—
233#131—2122no—
234C2×M16—531no—
235C2×M16—531no—
236C2×M16—1062no—
237C22×C8—1062no—
238C22×C8—1061no—
239C22×C8—1061no—
240C4⋊Q8—531no—
241C4⋊Q8—531no—
242C4⋊Q8C4×D5311yescharacteristic
243#127C4×D5311yescharacteristic
244#125—1061no—
245#124—21no—
246C22⋊Q8—1061no—
247C22⋊Q8—1061no—
248#121—21no—
249#121C22⋊D5312yes—
250C4×Q8—1061no—
251C4×Q8—1061no—
252C4×D4—21no—
253C4×D4D21212yes—
254#117—1061no—
255C2×C4⋊C4—21no—
256C2×C42—21no—
257C2×C42C22×D5311yescharacteristic
258#106—1062no—
259C4⋊C8—21no—
260C4⋊C8—531no—
261C4⋊C8—531no—
262C4⋊C8—1061no—
263#103—1062no—
264#102—1062no—
265C22⋊C8—21no—
266C22⋊C8—1062no—
267C22⋊C8—1062no—
268#97—1061no—
269C4×C8—21no—
270C4×C8—531no—
271C4×C8—531no—
272C4×C8—1061no—
273C4×C8—1061no—
274#95—21no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—1061no—
281C2×Q8—1061no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4#1171992112yes—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4D4×D5311yescharacteristic
289C22×C4#1171994711yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291SD16—2124no—
292M16—532no—
293M16—1061no—
294M16—1061no—
295M16—1061no—
296M16—2124no—
297C2×C8—41no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—1061no—
304C2×C8—2124no—
305C2×C8—2122no—
306C2×C8—2121no—
307C2×C8—2121no—
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—1061no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C4⋊C4#1171992112yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—2121no—
321C42—41no—
322C42—41no—
323C42—2121no—
324C42C2×C4×D5311yescharacteristic
325C42C2×D21211yescharacteristic
326C42D4×D5311yescharacteristic
327C42#1171994711yescharacteristic
328C23—22no—
329C23#1313318911yescharacteristic
330Q8—41no—
331Q8—41no—
332Q8—2122no—
333Q8—2121no—
334Q8—2121no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4C2×#1171992111yescharacteristic
354C2×C4#1313317711yescharacteristic
355C2×C4#1313317711yescharacteristic
356C2×C4#1313318711yescharacteristic
357C2×C4#1313318911yescharacteristic
358C2×C4#1313320111yescharacteristic
359C2×C4SD16×D5311yescharacteristic
360C2×C4#1313323611yescharacteristic
361C2×C4#1313324111yescharacteristic
362C2×C4#1313324611yescharacteristic
363C8—81no—
364C8—1062no—
365C8—1062no—
366C8—2121no—
367C8—2121no—
368C8—2121no—
369C8—2121no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1681050011yescharacteristic
376C22#1681082611yescharacteristic
377C22#1681083611yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—82no—
383C4—4241no—
384C4—4241no—
385C4#1681071411yescharacteristic
386C4#1681072011yescharacteristic
387C4#1681073711yescharacteristic
388C4#1681074211yescharacteristic
389C4#1681077511yescharacteristic
390C4#1681078611yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2027504011yescharacteristic
395C2#2027504711yescharacteristic
396C2#2027509611yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1502C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#1502D53
hypocenterC53#9055
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20275658C2semidirect
4. #20273119C2semidirect
5. #20272556C2semidirect
6. #20272551C2semidirect
10. #16811019C4semidirect
11. #16810995C4semidirect
12. #16810975C4semidirect
13. #16810969C4semidirect
118. #11719917SD16semidirect
229. C53#9055semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202852702
#202834592
#202756582
#202731192
#202725562
#202725512
C53×#15022
#905553